A Study on the p-Adic Integral Representation on Z p Associated with Bernstein and Bernoulli

doi:10.1155/2010/163217

Research Article

A Study on the p-Adic Integral Representation on Z p Associated with Bernstein and Bernoulli

Polynomials

Lee-Chae Jang,

Won-Joo Kim,

and Yilmaz Simsek

1Department of Mathematics and Computer Science, Konkuk University, Chungju 138-701, Republic of Korea

2The Research Institute of Natural Sciences, Konkuk University, Seoul 138-701, Republic of Korea

3Department of Mathematics, Faculty of Arts and Science, University of Akdeniz,Antalya, Turkey

Correspondence should be addressed to Lee-Chae Jang,[email protected] Received 13 August 2010; Accepted 15 September 2010

Academic Editor: Toka Diagana

Copyrightq2010 Lee-Chae Jang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the Bernstein polynomials on Zp and investigate some interesting properties of Bernstein polynomials related to Stirling numbers and Bernoulli numbers.

1. Introduction

LetC0,1denote the set of continuous function on0,1. Then, Bernstein operator forf ∈ C0,1is defined as

Bn

f

x ⁿ

k0

f k

n n

k

x^k1−x^n−kn

k0

f k

n

Bk,nx, 1.1

fork, n∈Z, whereBk,nx ⁿkx^k1−x^n−kis called Bernstein polynomial of degreen. Some researchers have studied the Bernstein polynomials in the area of approximation theorysee 1–6.

Let p be a fixed prime number. Throughout this paper Zp, Qp, C, and Cp will, respectively, denote the ring ofp-adic rational integers, the field ofp-adic rational numbers, the complex number field, and the completion of algebraic closure ofQp. LetUDZpbe the

set of uniformly differentiable function onZp. Forf∈UDZp, thep-adicq-integral onZpis defined by

Zp

fxdµx lim

N→ ∞

1 p^N

p^N−1 x0

fx 1.2

see4,7–15.

In the special case, if we setfx xⁿin1.2, we have

Bn

Zp

xⁿdµx. 1.3

In this paper, we consider Bernstein polynomials on Zp and we investigate some interesting properties of Bernstein polynomials related to Stirling numbers and Bernoulli numbers.

2. Bernstein Polynomials Related to Stirling Numbers and Bernoulli Numbers

In this section, forf∈UDZp, we consider Bernstein type operator onZpas follows:

Bn

f

x ⁿ

k0

f k

n n

k

x^k1−x^n−kn

k0

f k

n

Bkx, 2.1

forn∈ Z, whereBk,nx ⁿkx^k1−x^n−k is called Bernstein polynomial of degreen. We consider Newton’s forward difference operator as follows:

Δfx fx1−fx, Δⁿfx ⁿ

k0

n k

−1ⁿ−kfxk ⁿ

k0

n k

−1^kfxn−k. 2.2

Forx0,

Δⁿf0 ⁿ

k0

n k

−1^kfn−k ^∞

n0

n k

−1^n−kfk. 2.3

Then, we have

fn 1 Δⁿf0 ⁿ

l0

n l

Δ^lf0. 2.4

From2.4, we note that

fx ^∞

n0

x n

Δⁿf0, 2.5

where

Δⁿf0 ⁿ

k0

n k

−1^kfn−k. 2.6

The Stirling number of the first kind is defined by n

k1

1kz ⁿ

k0

S1n, kz^k, 2.7

and the Stirling number of the second kind is also defined by n

k1

1 1kz

ⁿ

k0

S2n, kz^k. 2.8

By2.5,2.6,2.7, and2.8, we see that

S2n, k 1 k!

k j0

−1^j k

j

k−j_n

, 2.9

whereΔⁿ0^{m n}_k0ⁿ_k−1^kn−k^m. Note that, fork∈Zandx∈0,1,

F^kt, x t^ke^1−xtx^k k! x^k

∞ n0

nk k

1−xⁿ t^nk nk!

^∞

nk

n k

x^k1−x^n−k tⁿ n! ^∞

n0

Bk,nxtⁿ n!.

2.10

Thus, we note thatt^ke^1−xtx^k/k! is the generating function of Bernstein polynomial. It is easy to show that

1 ⁿ_k

Zp

Bk,nxdµx ^n−k

l0

n−k l

−1^l

Zp

x^lkdµx ^n−k

l0

n−k l

−1^lBnk. 2.11 By2.11, we obtain the following theorem.

Theorem 2.1. Forn, k∈Zwithn≥k, one has 1

ⁿ_k

Zp

Bk,nxdµx ^∞

m0 n−k

l0

n−k l

−1^lBnk, 2.12

whereBnare thenth Bernoulli numbers.

In12, it is known that

xⁿⁿ

k0

x k

k!S2n, k, 2.13

n ki−1

_k

i

ⁿ_iBk,nx xⁱ, 2.14 fori∈N. By1.1and2.14, we see that

x^{i ∞}

m0

n−im−1 m

−1^mx^n−i−m1−x^{m n}

ki−1

_k

i

ⁿ_iBk,nx

^∞

m0

n ki−1

_k

i

ⁿ_i

n−im−1 m

n k

−1^mx^n−i−mk1−x^nm−k

^∞

m0

n ki

nm−k

l0

n−im−1 m

nm−k l

n k

×−1^lmx^ln−i−mk,

2.15

fori∈N. By2.15, we obtain the following theorem.

Theorem 2.2. Forn, k∈Z, andi∈N, one has

Bi^∞

m0

n ki

mn−k

l0

n−im−1 m

mn−k l

n k

−1^lmBln−i−mk. 2.16

From2.13and2.14, we note that n

ki−1

_k

i

ⁿ_iBk,nx ⁱ

k0

x k

k!S2i, k. 2.17

In16, it is known that

Zp

x n

dµx 1

n1. 2.18

By2.17,2.18, and Theorem2.2, we have

Bn^m

k0

k!

k1−1^kS2k, n−k. 2.19

From the definition of the Stirling numbers of the first kind, we drive that x

n

n! xnⁿ

k0

S1n, kx^k. 2.20

By2.17,2.19, and2.20, we obtain the following theorem.

Theorem 2.3. Fork, n∈Zandi∈N, one has n

ki−1

_k

i

ⁿ_iBk,nx ⁱ

k0

k l0

S1n, lS2i, kx^l. 2.21

By Theorems2.2and2.3, we obtain the following corollary.

Corollary 2.4. Fork∈N, one has

Bix ⁱ

k0

k l0

S1n, lS2i, kBl, 2.22

whereBiare theith Bernoulli numbers.

Acknowledgment

The present investigation was supported by the Scientific Research Project Administration of Akdeniz University.

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